Enfriamiento de estrellas de neutrones

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(1)Abstract The subject of Neutron Star Cooling is discussed. First, some general properties of neutron stars are treated in order to put the problem of Neutron star cooling in context. Cooling curves of isolated Neutron stars undergoing neutrino emission are calculated, and are finally compared with actual observed temperatures..

(2) Neutron Star Cooling Paul Nuñez Facultad de Ciencias Departamento de Fı́sica Universidad de los Andes Bogotá, Colombia January 16, 2006.

(3) Contents 1 Neutron Stars 1.1 The Origin of Neutron Stars . . 1.1.1 Type II Supernovae . . 1.2 Neutron Star Structure . . . . 1.2.1 Composition . . . . . . 1.3 The Equation of State . . . . . 1.3.1 The pure Neutron Star . 1.4 Neutron star stability . . . . . 1.4.1 Masses and Radii . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. 5 6 6 8 8 10 10 12 12. 2 Observations 2.1 Pulsars . . . . . . . . . . . . . . . . 2.1.1 Age determination . . . . . 2.2 Thermal emission . . . . . . . . . . 2.3 The Observed Temperature . . . . 2.3.1 The Temperature Redshift .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. . . . . .. 16 16 17 18 19 19. 3 Cooling 3.1 Cooling modes . . . . . . . . . . . . . . . 3.1.1 Photon Emission . . . . . . . . . . 3.1.2 Neutrino Emission . . . . . . . . . 3.2 The Direct Urca Threshhold . . . . . . . . 3.3 The Modified Urca Process . . . . . . . . 3.4 Cooling curves: An outline of the problem 3.4.1 Procedure . . . . . . . . . . . . . . 3.5 Direct Urca Luminosity . . . . . . . . . . 3.6 Modified Urca Luminosity . . . . . . . . . 3.7 Specific Heat Capacity . . . . . . . . . . . 3.8 Cooling Curves . . . . . . . . . . . . . . . 3.8.1 Direct Urca Cooling . . . . . . . . 3.8.2 Modified Urca Cooling . . . . . . . 3.9 Comparison to actual observations . . . . 3.10 Concluding Remarks . . . . . . . . . . . .. . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . .. 25 25 25 26 27 28 29 29 30 37 43 46 46 49 49 53. . . . . . . . .. 1.

(4) CONTENTS A The Oppenheimer-Volkoff Equation A.1 The spherically symmetric metric . . A.1.1 The Field Equation . . . . . A.2 Curvature . . . . . . . . . . . . . . . A.2.1 The Spin Connection . . . . . A.2.2 The Curvature Two-forms . . A.2.3 The Einstein tensor . . . . . A.3 The Energy-Momentum Tensor . . . A.4 Hydrostatic equilibrium . . . . . . . B The B.1 B.2 B.3. 2. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. . . . . . . . .. 55 55 55 56 56 57 59 59 59. Fermi Golden Rule 62 Time Dependent Perturbation Theory . . . . . . . . . . . . . . . . . . 62 Time independent perturbation . . . . . . . . . . . . . . . . . . . . . . 64 Transition to a continuum of states . . . . . . . . . . . . . . . . . . . . 64.

(5) Introduction We have come a long way since Neutron Stars were first postulated as theoretical constructs in the 1930’s. Ever since the first pulsar was observed in the 1960’s, N.S’s (Neutron stars) have been studied extensively and are still subject of research today. The very high densities present in N.S’s (∼ 10 14 g/cm3 ) and extreme conditions present are by far unreachable here on earth. Consequently, N.S.s are almost ideal astrophysical laboratories that we can use to study matter in extreme conditions and to test our current knowledge of physics. As we shall see, a beautiful mixture of particle physics, statistical mechanics and general relativity, among other fundamental fields, are used to describe N.S.s. The existence of Neutron stars was first proposed by Landau shortly after the Neutron was discovered (1931). The idea was then further refined by Baade and Zwicky in 1934, in their pioneering work on Supernovae. They made the accurate prediction that N.S.s were formed in Supernova explosions. Later, in 1938, the first N.S. model was made by Volkoff and Oppenheimer, a model being strongly based on General Relativity. For the next thirty years the idea of the neutron star was pretty much left aside on the basis that N.S. theory seemed to be a wild extrapolation of modern physics and also because it was thought that these objects would simply be too faint to be observed. It was not until 1967, when Jocelyn Bell and Anthony Hewish, discovered the first Pulsar (An object emmiting periodic radio signals). At first it was thought, that these periodic pulses came from extraterrestrial life. Soon a more convincing mechanism for the emission of these pulses was understood (Syncrotron radiation), and ever since, Neutron stars have been studied and observed extensively1 . Natural questions concerning N.S’s such as what are they made of (besides Neutrons, obviously), what are their dimensions, and how do they evolve are still being made today. Many plausible answers to these questions have been given, so all that can be done is try to discriminate dome of the answers given. How can this be done? Simple. We need to compare our theoretical models with observations. As we shall see, one of the most important observations that can be made on a N.S. is it’s temperature. 1. Historical facts were found in [2, 5, 4]. 3.

(6) CONTENTS. 4. The temperature-Time curve depends strongly on the type of matter and it’s structure, and the former determines the Equation of state. The E.O.S. (Equation of State), together with General Relativity give us the equilibrium conditions (maximum mass and radius). Consequently, by constructing cooling curves for various types of matter present 2 and comparing with observations one can not only constrain the E.O.S. and determine properties of the N.S., but also determine if exotic matter is actually present in the universe. The “core” of this document is presented in chapter III, but the importance of the previous two chapters lies in the fact that they put the problem of Neutron star cooling in context. Also, some important results will be derived in the first two chapters, which besides from being important to the problem in question, are fundamental results in the field of astrophysics. This document tries to be as self contained as possible, so that some of the equations presented in the document are derived and treated a bit more deeply in the appendixes.. 2. Among the various types of matter considered are: Ordinary matter, quark matter, Pion condensates, Kaon condensates, etc..

(7) Chapter 1. Neutron Stars Main sequence stars produce energy by fussion, starting from hydrogen, and progressing towards heavier elements. When 56 F e is reached, it is no longer energetically favorable to produce heavier elements, so gravity starts to take over until the counter-pressure produced by Pauli repulsion in the electrons may leave the star in equilibrium. This new equilibrium state is commonly called a White Dwarf. However, if the mass of the star is sufficiently large (> 1.4 M , M being the solar mass), the zero point pressure (produced by the Pauli exclusion principle) is no longer capable of supporting the star against gravitational collapse. A massive implossion and then an explosion, called a Supernova, takes place. The remnant of this spectacular event is what is now known as a Neutron Star. This remnant Neutron star has a mass of ∼ 1.5M and a radius of the order of ∼ 10km [1, 4, 2], thus corresponding to densities of the order of 10 14 g/cm3 . These densities correspond to energies much greater than the threshold energy for electron capture, so we expect that the neutron star is composed primarily of neutrons. The counter-pressure that keeps the N.S. from collapsing is the zero point pressure for a degenerate gas of neutrons. Just as in the case of White Dwarfs we expect there to be a limiting mass, and since a degenerate gas of neutrons might be the densest form of matter, the outcome of a possible collapse is a Black Hole. Neutron Stars may emmit periodic radio signals and are thus called Pulsars(See chapter II). Currently, more than 1000 pulsars have been detected [5]. Neutron stars also emmit thermal radiation in the X-ray frequency range, and when in a binary accreting system, they also emmit radiation in the X-ray region. Neutrinos, that are virtually undetectable1 , are emmited in abundance. These neutrinos, as we shall see, are crucial in the cooling process. In the following subsections, some properties of neutron stars will be discussed in more detail, and the problem of neutron star cooling will be put in context. 1 It is hard enough to detect neutrinos from the sun. Recall that neutrinos only interact via the weak interaction.. 5.

(8) CHAPTER 1. NEUTRON STARS. 1.1. 6. The Origin of Neutron Stars. We shall now discuss the case in which the mass of the initial main sequence star is greater than the limiting Chandrasekar 2 mass of 1.4M . We shall briefly describe the Supernova event.. 1.1.1. Type II Supernovae. Neutron Stars are formed in type II Supernovae 3 . This event only lasts a few seconds after nuclear fussion has completely halted. Since the electron pressure is not enough to keep the star from collapsing, an implosion occurs. Soon, densities greater than 8 × 106 g/cm3 begin to appear [4]. This is the minimum density for electron capture to take place (p + e → n + ν). Since electron capture is the key mechanism responsible for the Neutron Star formation, it might serve us to calculate the minimum density for which this process occurs. The minimum energy required for be found by simply saying that. 56 F e. to capture energetic electrons capture can. M (A = 56, Z = 26)c2 + Emin = M (A = 56, Z = 25)c2 ,. (1.1). giving Emin = 1.88 × 10−13 J = 1.803M eV. It is now plausible to say that this minimum energy is mostly supplied by the degenerate electrons with Fermi energy 4 Emin = Ef (e) Ef (e) =. . h̄ Ne 3π 2 2me V. 2/3. = 1.803M eV .. (1.2). Noting that Ne = Np because of charge conservation, and that the matter density can be expressed as ρmin. Np = V. . . A Ne mB = Z V. . . A mB Z. (1.3). A Were Z is the number of barions per electron, and m B is the mass of a barion. In the case of 56 F e, A Z = 2.15 [5]. We can now rewrite eq. ( 1.2) as 2. Chandrasekar predicted the maximum mass for a white dwarf by modeling it as a degenerate gas of electrons. 3 In type I supernovae, there is initially a White Dwarf star that accretes mass from its binary companion. When the white Dwarf has accreted enough mass from its companion, a carbon-oxygen core is ignited and a shock wave is produced that destroys the star, there is no Neutron Star left [6]. 4 Eq. 1.2 is a known result for the energy of a gas of fermions at T = 0.

(9) CHAPTER 1. NEUTRON STARS. 7. h̄2 Ef (e) = 2me. 3π 2 Z mB A. !2/3. 2/3. ρmin .. (1.4). So . . Ef (e)2me 3/2 mp A = 8.64 × 106 g/cm3 , ρmin = 3π 3 Z h̄2 which is consistent with what is mentioned by Baym [4].. (1.5). This density is still much less than the Nuclear density, this being ρ 0 = 3 × [4], consequently, there is a massive neutronization and neutrino production. The collapse halts when the density approximately reaches ρ 0 , due to the high incompressibility of nuclear matter. The resistance to the collapse of this very dense core is in part responsible for a shock wave that turns the implosion into an explosion [6]. For an instant, the high opacities inside the star, make it impossible for the produced neutrinos to escape. Further neutrino production is momentarily suppressed because a degenerate gas of neutrinos is momentarily formed, i.e. for a neutrino to be produced, it would have to occupy a state that is probably already occupied. 1014 g/cm3. It is thought that Neutrinos play a crucial role in the final ejection of the stellar mantle [1, 6]. Once the trapped neutrinos escape and thus, the stellar mantle is ejected, there is “room” (In phase space terms) for further neutronization to occur. Note that further neutronization causes the reduction of the number of electrons per unit volume, and thus a further decrease in the electron or zero point pressure 5 . Only when the concentration of neutrons becomes high enough, the star remnant can reach equilibrium because of the zero point pressure produced by the newly formed neutrons. What is now left is a star composed mainly of neutrons (hence the name of the star) with an initial temperature of T i ∼ 1011 ◦ K [3, 1]. It is now illuminating to calculate the Fermi energy of a gas of neutrons at nuclear density. Using eq.( 1.2) we find Ef (n) ≈ 1.38 × 10−11 J = 86.13M eV . This corresponds to a Fermi temperature of 1012 ◦ K. The temperature of the initial neutron star is roughly one order of magnitude less than the Fermi tempereture, so the N.S. can be regarded as a degenerate neutron gas (See figure 1.1.1). The fact that the initial N.S. is degenerate can be seen in figure . Note that as the temperature drops, the N.S. will become even more degenerate . This is the reason why a neutron star can be thought as a cold star. It is important to restate that the N.S. now supports itself against gravitational collapse by the neutron zero point pressure. The problem of N.S. cooling will be treated with these initial conditions. As a final remark, it is worth it to say that the subject of Supernovae is by no means fully understood and is still subject to research today. 5. Ef (e) ∝. Ne V. 2/3. and Pe ∝. Ne V. 4/3.

(10) CHAPTER 1. NEUTRON STARS. 8. Neutron Star at Birth 1.2. T=1e10 Kelvin T=1e11 Kelvin. 1. f (Occupation probability). 0.8. 0.6. 0.4. 0.2. 0. 0. 5e-12. 1e-11. 1.5e-11. 2e-11. 2.5e-11. E (Joules). Figure 1.1: The Fermi distribution is plotted for an initial temperature of 10 11 ◦ K, and then for a temperature of 1010 ◦ K, as we shall see, the former temperature is reached in only a few hours!. Note that almost all the energy levels beneath the Fermi energy are occupied. As the temperature keeps dropping, the N.S. will become even more degenerate.. 1.2. Neutron Star Structure. The process of N.S. formation gives us a rather good idea about the global structure of a N.S. . Although a N.S. is composed mainly of neutrons, heavy nuclei also exist in the crust and some protons and electrons exist in the core along with the neutrons. First we shall discuss the global structure in more detail. Then, the equation of state, and the limiting masses of the N.S. will also be treated.. 1.2.1. Composition. A cross section of a tipical N.S. is shown in figure ( 1.2). In the surface one expects there to be an abundance of 56 F e, electrons are degenerate just as they are in a White dwarf star. As one goes further inside, the threshhold density for electron capure is reached6 (ρmin = 8 × 106 ), so nuclei become more and more neutron rich, reaching nuclei such as 118 Kr at a density ρd = 4.3 × 1011 g/cm3 [4, 2]. These nuclei, although unstable in a laboratory, do not present β decay (n → e + p + ν) because there is still a degenerate electron fluid. In other words, if a heavy nucleus such as 118 Kr, were to present β decay, the electron would have to go to an energy state that would probably be occupied. The outer region of the neutron star containing nuclei is called the Crust [4, 2]. 6. See subsection 1.1.1.

(11) CHAPTER 1. NEUTRON STARS. 9. Figure 1.2: Schematic cross-section of a N.S.. Picture taken from the article of Baym [4] The density ρd = 4.3 × 1011 g/cm3 is commonly called the neutron drip density. At densities greater than ρd , nuclear forces begin to saturate, as a consequence of them being short range. What happens now is that neutrons begin to “drip” out of the nuclei. As neutrons begin to drip out, unusual structures begin to appear. The shape of nuclei changes from spherical to rod like, to laminar. These unusual structures are called Pasta Nuclei [4]. Then, as densities keep increasing, and reach nuclear density (ρ0 = 3 × 1014 g/cm3 ), matter dissolves into a uniform liquid that is primarily composed of degenerate neutrons with a few protons and electrons. The inner region, after the crust, is called the Core. As we shall see, the amount of protons inside the N.S. will slightly vary the equation of state and might have dramatic effects on the way the N.S. cools. There are still uncertainties about the type of matter that exists at even higher densities. Pion or Kaon condensation may occur and form a superfluid Bose-Einstein condensate. Although superfluidity will not be treated in detail, it might also have strong effects on N.S. cooling. At ultrahigh densities, it is thought that free quarks might exist [4]. The core of the N.S. constitutes about 90% of the total volume. There is also an atmosphere composed of hydrogen that has no notable contribution on the total mass of the star, but plays an important role in shaping the emitted photon spectrum [1]..

(12) CHAPTER 1. NEUTRON STARS. 1.3. 10. The Equation of State. In this section, we shall assume, as a first (and reasonable) approximation, that the N.S. is composed entirely of neutrons. This is a valid approximation because the core, which is composed primarily of neutrons constitutes 99% of the mass of the Neutron star [1]. We shall also assume, that the neutrons are in the ground state. This is valid to say because, the neutrons are highly degenerate 7 . One final and crucial assumption, is to take the neutrons as non interacting 8 . The main reference for this section is Reddy’s article [5].. 1.3.1. The pure Neutron Star. To obtain the energy of the ground state of a pure neutron gas, we can proceed as follows [5]. U =2. |~ p|=pf. X q. p~2 c2 + m2 c4. (1.6). |~ p|=0. Here the sum is made over the neutron momentum. The factor of 2 just takes into account the spin of the neutron and p f is the Fermi momentum. We may now use the density of states (V /h3 ) and write eq. ( 1.6) as an integral. U=. 2V h3. Z. p=pf p=0. q. p~2 c2 + m2 c4 ; d3 p~.. (1.7). Now changing to spherical coordinates, the previous eq gives U=. 8πV h3. Z. q. |p|=pf |p|=0. p2 c2 + m2 c4 p2 dp.. (1.8). Solving this eq might not be as illuminating as looking at the relativistic and non relativistic limits of eq. ( 1.8). First lets look at the non relativistic limit (making a series expansion for small momenta). U=. 8πV p5f h3 10mn. (1.9). We now need a relation between pf and V given by 2V N= 3 h 7. Z. pf 0. 8πV p3f d p~ = , 3h3 3. (1.10). See subsection 1.1.1 If interactions are taken into account, the bare mass of the neutron should be replaced by the effective mass. 8.

(13) CHAPTER 1. NEUTRON STARS. 11. So that 3N h3 8πV. pf =. !1/3. .. (1.11). Eq. ( 1.9) becomes 8π U= 10mn h3. 3N h3 8π. !5/3. We can now calculate the pressure using P = −. Pnon−rel =. 8π 15mn. 3N h3 8πV. =. 8π 15mn. 3h3 8πmn. V −2/3 . . ∂U ∂V. . T =0. (1.12) , so that. !5/3. !5/3. (1.13) ρ5/3. (1.14). Note that P ∝ ρ5/3 . This is commonly called a polytropic equation of state with γ = 5/3. Next we calculate the equation of state for the very relativistic case. 8πV U= 3 h ⇒ Prel. 2πc = 3 3h. Z. pf 0. 2πc p c dp = 3 h 3. 3N h3 8πV. !4/3. 2πc = 3 3h. 3N h3 8πV. !4/3. 3h3 8πmn. !4/3. (1.15). ρ4/3 .. (1.16). This is a polytrope with γ = 4/3 . The non relativistic case corresponds to low pressures and low densities. In the relativistic case, neutrons are taken to be highly energetic and thus the approximation corresponds to high pressures and high densities within the star. More generally, one can express the equation of state as [5] P = An−r ρ5/3 + Ar ρ4/3 .. (1.17). These results are of course very approximate because we have not taken into account nucleon-nucleon interactions, a few more sophisticated models have been proposed that include interactions. The subject of the equation of state is still subject to research today9 . 9. See Shapiro [2].

(14) CHAPTER 1. NEUTRON STARS. 12. The methods used to calculate the equations of state, apply just as well for White dwarfs, but instead of there being a degenerate gas of neutrons, one takes there to be a degenerate gas of electrons. As we shall see in the next section, the equation of state is crucial in determining the maximum and minimum masses of a neutron star.. 1.4. Neutron star stability. In section 1.1.1 we concluded that the equation of state could be expressed in two limits: A non relativistic limit (eq. 1.14) which corresponds physically to low pressures and densities, and a very relativistic limit (eq. 1.13) that corresponds to extremely high pressures and densities. In previous sections (see 1.2.1) we have also stated that there is a density at which neutrons start to drip out of nuclei ρ d . This density can be thought of as the limit of unstable white dwarfs because it corresponds to the transition from heavy nuclei to a neutron liquid. Similarly, we shall see that there is a maximum density allowed, which is thought to be ρmax = 5×1015 g/cm3 (Pretty close to nuclear density.). We shall now present the famous equations for stellar equilibrium, and along with the densities mentioned in the preceding paragraph and the equations of state, we can calculate the limiting masses and radii for Neutron stars.. 1.4.1. Masses and Radii. In a normal main sequence star of 1M , the Shwartzschild radius (2GM/c2 ) is of the order of a few kilometers and is thus negligible to the radius of the star. This is why Newtonian gravity can be used to describe the main sequence star 10 . On the other hand, a typical N.S. will have a radius of a 10 or 12km, which is comparable to its Swartzchild radius, so a General Relativistic description of gravity must be used. The equations used to describe the hydrostatic equilibrium of a General Relativistic star are the following pair of coupled differential equations. . −Gm(r) ρ(r) dp(r) p(r) = 1+ dr r2 ρ(r)c2. . 1 + 4πr 3 p(r) 1+ m(r)c2. !. 1+. 2Gm(r) c2 r. −1. (1.18). dm(r) = 4πr 2 ρ(r) (1.19) dr These are commonly known as the Tolman-Oppenheimer-Volkov (O.V.) equations, and have a purely General relativistic origin 11 . Now that we know the equation of state p(ρ), and values for central densities (ρ max , ρmin ), we can solve this system 10 11. If the star were to shrink beyond this radius, it would collapse into a Black Hole. See Appendix A for a derivation of these equations..

(15) CHAPTER 1. NEUTRON STARS. 13. of differential equations (usually done numerically). What one obtains is a solution for m(r) and ρ(r). The radius ’r’ at which the density drops to zero, is the radius of the star. Now with the O.V. equations and the equation of state we can construct a “Stellar model”. As an example of an application of the O.V. equations, we can calculate the minimum mass at which a Neutron star could possibly be stable. If one solves the O.V. equations taking the neutron drip density ρ d as the minimum density and eq. ( 1.14) as the equation of state, one obtains what is shown in figure 1.3. One obtains the minimum stable mass of a N.S. to be around 0.03M , which is consistent with that obtained in the literature (see Shapiro [2]). This result is however a bit unrealistic in the sense that supernova remnants usually have masses larger than this. The minimum mass of a supernova remnant is usually of the order of 1M [1], which is 12 actually closer to the maximum allowed mass of a White Dwarf . To find the radius and density corresponding to the alleged minimum mass of 1M , we can vary the central density and solve the O.V. equations. A mass of 1M is found at a central density of ρ c = 1 × 1014 g/cm3 using a relativistic equation of state (Eq. 1.13). This “Low density” star is found to have a radius of ≈ 91km. Now if one starts to increase the central density, and constructs a model of the form of figure 1.3, using a relativistic eq of state of the form 1.13, one obtains a maximum allowed mass of the order of 1.75M as shown in figure 1.4. This maximum mass corresponds to a central density of the order of 5 × 10 15 g/cm3 and is consistent with what is mentioned by Lattimer [1]. It is natural to expect a maximum mass corresponding to a density a few times greater than nuclear density, since it is thought that no fundamental force can support densities greater than 5 × 10 15 g/cm3 . The results of this section will be used in later chapters and can be summarized as: 1M. < M < 1.75M. (1.20). 12km < R < 91km. (1.21). 1 × 1014 g/cm3 < ρc < 5 × 1015 g/cm3. (1.22). These results of course vary (Drastically in some cases.) if one uses more sophisticated equations of state that take into account nucleon interactions, for this reason, there is still no agreement on the limiting masses (especially on the maximum mass). Nevertheless, these results are good enough for our present purposes. 12. Mmax ∼ 1.44M . This is the famous Chandrasekhar result..

(16) CHAPTER 1. NEUTRON STARS. Density. 14. Limiting Mass. 14. 4·10. 14. 3·10. 14. 2·10. 14. 1·10. 10000 M. 20000. 30000. 40000. 50000. 40000. 50000. R. Limiting Mass. 0.02. 0.015. 0.01. 0.005. 10000. 20000. 30000. R. Figure 1.3: Stellar model for a low density neutron star (corresponding to an eq. of state of the form of eq. 1.14). The radius is measured in meters, the mass in solar masses and the density in [g/cm3 ]. Note that the density goes to zero at the radius of the star. The minimum mass can be calculated by evaluating m(r) were the density goes to zero. Note that the minimum mass is of the order of 0.03 M ..

(17) CHAPTER 1. NEUTRON STARS. M. 15. Limiting Mass. 1.75 1.5 1.25 1 0.75 0.5 0.25 10000. 20000. 30000. R. Density 18. 2.5·10. 18. 2·10. 18. 1.5·10. 18. 1·10. 17. 5·10. 10000. 20000. 30000. R. Figure 1.4: Model for a high density N.S., taking the central density to be 5 × 1015 g/cm3 . The density goes to zero at the radius of the star. The maximum mass can be calculated by evaluating m(r) were the density goes to zero. M max ≈ 1.75M ..

(18) Chapter 2. Observations Neutron Stars were first observed as Pulsars, which are objects emmiting periodic radio signals, today, more than 1000 pulsars have been detected. In this chapter, we shall briefly describe thermal and non thermal related radiation emission. Mass measurements will not be discussed; still, it is worth saying that these measurements are obtained with the help of Kepler’s third law observing N.S. binaries. The inferred masses are consistent with the masses obtained in the previous chapter, and thus with our knowledge of late stellar evolution [2]. In this chapter we shall be discussing the observations made on isolated N.S.s, which can have masses and radii in the ranges calculated in the previous chapter.. 2.1. Pulsars. The high spin rates1 that N.S. possess give rise to extremely high magnetic fields of the order of 1012 G, the magnetic axis usually makes an angle with the rotation axis as shown in figure ( 2.1). The rotating magnetic field thus produces an electric field (By Faraday’s induction law.) that ejects electrons from the surface. These ejected electrons will follow helicoidal paths along the magnetic flux lines and will thus emmit radiation as shown in figure ( 2.1). Now, if by any chance, an observer is momentarily colinear with the magnetic axis, he (or she) will observe periodic light signals, typically ranging from milliseconds to seconds. This radiation is usually in the broad-band radio range. What we just briefly described is called the pulsar mechanism, and is also known as the Lighthouse model [12]. The pulsar depicted in figure 2.1 has a relatively narrow radiation beam, of only a few degrees across [12]; as a consequence of this, not all N.S.s can be observed as pulsars. The radio pulses described can be detected by radio telescopes here on earth, and were first observed in 1967 by Jocelyn Bell, a graduate student; 1. Since the radius of the star decreases during the Supernova event, conservation of angular momentum causes the newly born N.S. to have very high spin rates, of the order or 1 full revolution per second.. 16.

(19) CHAPTER 2. OBSERVATIONS. 17. Figure 2.1: A schematic representation of a Pulsar. Taken from Astronomy Today. [12] their interpretation however, was given by her thesis advisor, Anthony Hewish. It is important to emphasize that this radiation is not thermally related, and is not emitted directly from the surface. Still, pulsars are worth being mentioned because once such an object is discovered, one is almost certain to have found a N.S., and thus, more observations can be made. In particular, once a pulsar is found, thermal observations can be performed.. 2.1.1. Age determination. The age of N.S.s can only be known exactly if the pulsar is related to a historical supernova. For example, the Chinese documented in the year 1054, the first supernova. Today this supernova remnant is known as the Crab Pulsar. Kepler and T. Brahe also documented supernovae in 1604 and 1572 respectively, however, these are not documented pulsars. There are 7 historical supernovae that have occurred in this galaxy, but the only one were we can be sure that a N.S. exists is the Crab Pulsar because, as it’s name suggests, it is a documented pulsar [2]. In case matter ejected from a supernova is detected, one could in principle measure its velocity. If the veocity can be measured (not always) one could calculate the time at which all matter was at the same place, and thus calculate the age of.

(20) CHAPTER 2. OBSERVATIONS. 18. the supernova remnant, which is in many cases, a N.S.. This has actually been done for the Crab and Vela pulsars, and the age determined for the Crab pulsar coincides with what was documented by the Chineese [12]. For the rest of the observed pulsars, the age cannot be determined exactly. Still, a caracteristic time can be associated by knowing the period P and the period-time variation Ṗ . A characteristic time τ can be defined by noticing that dP dt ∼ P τ. (2.1). so that P (2.2) Ṗ The estimation of a pulsar age was given by Gunn and Ostriker (1969), with the help of their magnetic dipole model for pulsars [2]. They found that to a good approximation, the age of a pulsar can be written as τ≡. P τ = (2.3) 2 2Ṗ The importace of the previous expression is that it only depends on measurable variables, in particular, the period of a pulsar can be measured with a precission of up to 14 significnt figures [2]. The previous equation gives a good estimation of the known age of the Crab Pulsar (A discrepancy of 325 years [2]). t≈. 2.2. Thermal emission. Thermal emission differs from non-thermal emission in the sense that the former originates at the surface of the N.S.. The wave length of the thermal light we perceive from N.S.s is in the X-Ray region. A rough estimation of the temperature using E = kB T = hc/λ yields a temperature of the order of ∼ M K. Observations of thermal emission became possible after the launch of the Einstein satelite (1979) and EXOSTAT satellite (1983) [13], which are equipped with X-ray telescopes. The latest X-ray observations were made by the Chandra satellite. There are several forms of surface emission. Perhaps the strongest signal is emitted by the magnetic poles of the N.S.. The accelerated particles create “hot spots” near the magnetic poles like the ones depicted in figure 2.1. These hot spots emmit radiation in the x-ray range, however, this radiation is considered non-thermal because it is related to accelerated charged particles. One can then observe x-ray pulsars. With the Chandra satelite, one can not only observe x-ray pulsars due to the magnetic poles, but also small pulses due to tiny variations in the surface temperature [13]..

(21) CHAPTER 2. OBSERVATIONS. 19. Taking the spectrum of the surface of a N.S. is not an easy task because nonthermal radiation is so bright that thermal radiation is hardly visible [13]. This is specially true with young N.S.s, with ages of the order of 10 3 yr. Also, the ejected material from the supernova can be hot enough for x-rays to be emitted, these can shadow the ones being emitted from the N.S. itself. Once the spectrum of the surface is taken 2 , there are a few ways of finding the surface temperature. One is making a black-body fit of the spectrum, and associating a temperature to the fitted curve. If the curve cannot be fitted using a black-body curve, it can be fitted assuming that the atmosphere of the N.S. is composed mainly of hydrogen (Hydrogen atmosphere model.) [13]. The lack of spectral absorption lines corresponding to heavy elements or even Helium suggests that the Hydrogen atmosphere model is plausible; also, strongly magnetized hydrogen does not seem to have visible spectral lines in the x-ray region [13]. The obtained temperatures for N.S.s are shown in figure 2.2. For the purpose of this document, we would prefer cooling curves, but temperatures alone are hard enough to obtain. Not all objects in figure 2.2 are documented pulsars (The ideal case); in fact, most are “x-ray point sources” in the middle of supernova remnants. The former statement poses doubts as to wether some of the “x-ray point sources” are indeed N.S.s. In the next section we shall discuss the relation between the observed temperature T∞ and the actual surface temperature T 0 .. 2.3. The Observed Temperature. Recalling the discussion of section 1.4.1, which implies that General Relativistic effects become non-negligible, one can conclude something very important about the observed temperature. As we shall see, the observed temperature here on earth is actually less than the temperature one would observe at the surface of the N.S.. For the following discussion, some knowledge of General Relativity is required, the result final result of the “temperature redshift” is given in eq. ( 2.27) and (2.26).. 2.3.1. The Temperature Redshift. If we neglect the effects of rotation, the geometry of space-time near the N.S. radius can be described by the spherically symmetric Schwarzschild metric 3 . This mertic (gµν ) can be written as [15] . 2GM ds = − 1 − r 2. 2. . . 2GM dt + 1 − r 2. −1. dr 2 + r 2 dΩ2. (2.4). A spectrum may be taken by graphing number of photons detected for a definite energy or wavelength. 3 To neglect rotation is a very bold thing to say because of the high spin rates that N.S.s possess. Rotation could be taken into account using a more realistic Kerr metric, but the algebra is simply too long..

(22) CHAPTER 2. OBSERVATIONS Source PSR J0205+5449 PSR B0531+21 (Crab) RX J0822-4330 1E 1207-52 PSR 0833-45 (Vela) PSR B0656+14 PSR 0633+1748(Gemina) RX J1856-3754 PSR 1055-52 Tycho Kepler SN 1006 RCW 86 W28. t [kyr] .82 1 2-5 ≤7 11-25 ∼ 110 ∼ 340 ∼ 500 ∼ 530 407 375 973 1794 3400. 20 T∞ [M K] < 1.1 < 2.1 1.6-1.9 1.1-1.5 0.65-0.71 0.91 ± 0.05 5.6 (+0.7, −0.9) 0.52 ± 0.07 0.82 (+0.06, −0.08) < 1.8 < 2.1 < 0.8 < 1.5 < 1.8. Figure 2.2: These temperatures were obtained by the Einstein and Chandra observatories, and were found in Shapiro’s book [2], and in Yakovlev’s article [14]. T ∞ refers to the temperature measured by an observer on earth (see next section). The ages that are known exactly correspond to historical supernovae. The uncertainties are basically due to interstellar absorption. were dt,dr and dΩ are one-forms, and ds is the differential of the invariant spacetime interval. Here we are using units were c = h̄ = 1 We are interested in null geodesics, which are the paths describe by free photons. Recall that a geodesic (Shortest-distance path) is a generalization of a straight line in a curved space-time. To find the shortest-distance path x µ (λ) in flat space time 4 , it is natural to simply write d dxµ = 0, (2.5) dλ dλ which is simply the equation of a straight line. A generalization of the previous equation to curved space-time can be written as D dxµ = 0, dλ dλ were 4. D dλ. (2.6). is called the directional covariant derivative, and is defined as. Here, λ is an appropriate parameter, and µ refers to the component of the four-vector. The µ = 0 component refers to time and the rest refer to the spacial components..

(23) CHAPTER 2. OBSERVATIONS. 21. D dxµ ≡ ∇µ (2.7) dλ dλ Recall that ∇µ is the covariant derivative. The covariant derivative of a vector µ x can be expressed as ∇µ xmu = ∂µ xµ + Γνµλ xλ .. (2.8). In the previous equation we are using the Einstein summation convention. The covariant derivative may be thought as a partial derivative ∂ µ = ∂x∂ µ plus a correction specified by a set of n matrices (n being the dimension of the manifold, in this case 4) Γνµλ , one matrix for each direction µ. The Γs are known as the Christoffel connection and are derived from the metric, these vanish in flat space-time as expected [15]. The geodesic equation ( 2.6) can now be written as σ ρ d2 xµ µ dx dx + Γ =0 (2.9) ρσ dλ2 dλ dλ Now lets look at one of the geodesic equations, namely the µ = 0 component. The only surviving connection coefficient is Γ 001 , so that we can write 1 0 d2 x0 0 dx dx + Γ = 0, 01 dλ2 dλ dλ. (2.10). or equivalently. The Γttr. d2 t dr dt + Γttr = 0, 2 dλ dλ dλ can be calculated directly from the metric g µν using [15] 1 Γσµν = g σρ (∂µ gνρ + ∂ν gρµ − ∂ρ gµν ). 2. (2.11). (2.12). So that Γttr = = = =. 1 tρ g (∂t grρ + ∂r gρt − ∂ρ gtr ) 2 1 tt g ∂r gtt 2 1 2GM −1 2GM 1− 2 r r2 GM . r(r − 2GM ). Now eq. ( 2.11) can be written as. (2.13).

(24) CHAPTER 2. OBSERVATIONS. 22. dr dt d2 t 2GM + = 0. 2 dλ r(r − 2GM ) dλ dλ. (2.14). dt . We can use the metric to We would like to have a differential equation for dλ dr dr obtain an expression for dλ in terms of dλ . By making ds = 0 in the Schwarzschild metric ( 2.4)5 , and also setting dΩ = 0, which implies only radial paths, on obtains. . 2GM 1− r. . . dt2 2GM = 1− 2 dλ r . −1. d2 r dλ2. (2.15). . dt 2GM −1 dr = 1− . (2.16) dλ r dλ With the previous equation in mind we can write the geodesic equation 2.14 as 2GM d2 t + 2 dλ r2 As we shall see, we are interested in. . dt dλ ,. dt dλ. 2. = 0.. (2.17). rather than t(λ). One can verify that. ω0 dt = 2 dλ 2 1 − 2GM r. (2.18). is a solution to eq. ( 2.17), were ω0 is a constant. The result we have just derived is important because it is the only non-zero component of the four-momentum µ 6 pµ = dx dλ as measured by a comoving observer . We are interested in the energy of the photon which is given by [15] E = −pµ uµ ,. (2.19). uµ uν = gµν uµ uν = g00 u0 u0 = −1.. (2.20). were uµ is the four-velocity of the comoving observer (At fixed spatial coordinates.). This observer would have a four-velocity such that u i = 0 for the spatial components and u0 is given by the normalization condition. The previous equation implies that the time component gives t. u =. √. −gtt =. s. 1−. 2GM . r. (2.21). We can now calculate the photon’s energy using equations ( 2.19) and ( 2.18), so that √ Recall that the proper time τ is related to the space-time interval as dτ = −ds2 , so making ds = 0 is equivalent to setting dτ = 0, which is always true for a photon. dx 6 Recall that one can always find an appropriate parameter λ for a null path such that pµ = dλµ is the four-momentum [15] 5.

(25) CHAPTER 2. OBSERVATIONS. 23. −gµν pν uν = −g00 p0 u0 ω0 ⇒ E= q . 2 1 − 2GM r. E. =. (2.22). Thus, if a photon is emmited from the N.S. surface at a radius R, it will have an energy ω0 E0 = q , 2 1 − 2GM R. (2.23). as measured by a comoving observer. When the photon has traveled a very long distance, it will have an energy given by E∞ = lim. r→∞. ω0. q. 2 1−. The previous equation implies that. 2GM R. s. E∞ = E 0 1 −. =. ω0 . 2. 2GM , Rc2. (2.24). (2.25). in S.I units. The previous equation implies that the photon has been gravitationally redshifted. We can now estimate the temperature as measured by a far away observer (T ∞ ), in terms of the surface temperature (T 0 ). E. ≈. kB T. (2.26) ω0. ⇒ To = q 2 1− s. 2GM k Rc2 B. ⇒ T∞ = T0 1 −. 2GM Rc2. The previous result is known as the Temperature Redshift, in the sense that the temperatures we can measure here on earth are actually less than the ones at the surface q of the N.S., we have just proved what is mentioned in Yakovlev’s article [14]. The 1 − 2GM factor is known as the Redshift factor, and can vary depending on Rc2 the N.S. mass and radius. Using the limiting masses and radii that were calculated in Chapter I (Eqs. 1.20- 1.21) we can conclude that 0.75 <. s. 1−. 2GM < 0.98. Rc2. (2.27).

(26) CHAPTER 2. OBSERVATIONS. 24. Surface Temperatures. log(T) (Kelvin). 1e+07. "temperaturas.txt". 1e+06. 100000 100. 1000. 10000 log(t) (yrs). 100000. 1e+06. Figure 2.3: Here we are plotting actual surface temperatures (T 0 ), which differ from the observed temperatures (T∞ ) by the redshift factor (eq. 2.27). This factor will play an important role in the next chapter because it will give us an uncertainty in the surface temperature. This uncertainty is actually much greater than the one related to observational error or uncertainty in interstellar absorption. It is important to emphasize that this uncertainty in the temperature is due to the uncertainty in the masses of the observed N.S.s. We can now plot the actual surface temperatures (T0 ) using what we have just learned (figure 2.3).

(27) Chapter 3. Cooling The origin of N.S.s as supernova remnants gives us a good idea of their global structure. Nontheless, there are still uncertainties on the matter composition of this astrophysical object. Different matter compositions will alter the equation of state and would thus alter some of the global properties treated in chapter I. The behavior of matter at extremely high densities is still not completely understood, and thus, questions such as what is the proton concentration in a N.S., and What kind of matter actually exists in the center of the N.S. could potentially be answered by studying the cooling mechanisms. As we shall see, the cooling of a N.S. is very sensitive to the type of matter present. The study of N.S. cooling provides us a way of looking directly inside the N.S., and to put to test much of our current knowledge of matter. In this chapter we shall discuss the two most favorite cooling mechanisms via neutrino emission, and the more inefficient cooling mode due to photon emission (Black body radiation). Finally, we will compare our theoretical cooling curves with observed temperatures.. 3.1. Cooling modes. 3.1.1. Photon Emission. The first cooling mechanism that comes to mind is black body radiation, one that we are more accustomed to; after all, many “earthly” objects cool this way. The energy lost via this mechanism can be expressed by using the Stephan-Boltzmann law: E˙γ = 4πR2 σT 4 ,. (3.1). were the Temperature T refers to the surface temperature of the outer crust mainly composed of heavy nuclei1 and σ is the Stephan-Boltzmann constant. Note that the energy loss via photon emission is proportional to R 2 . As we saw in section 1.4.1, the radius of a N.S. is of the order of 10km, which is very small compared 1. See section 1.2.1. 25.

(28) CHAPTER 3. COOLING. 26. for example to the radius of a White Dwarf star 2 . For this reason, the emission of photons is not an efficient cooling mechanism, and although a N.S. has very high temperatures, it may have a very low photon luminosity.. 3.1.2. Neutrino Emission. The idea that N.S.’s might cool via neutrino emission was first stated by Gamow and Schoenberg (1941), at about the same time that the importance of neutronization in supernovae was considered [3]. The simplest neutrino emmiting processes are beta decay of the neutron and electron capture of electrons: n → p + e + νe. (3.2). p + e → n + νe .. (3.3). Both of these reactions would have to have the same rate if the system is in chemical equilibrium. That is, if it satisfies the condition µn = µ p + µ e ,. (3.4). and thus, neutrinos can be produced continously. Reactions of the type 3.2, 3.3 are possible via the weak interaction as shown in figure 3.1.2. Cooling mechanisms that emmit neutrinos are commonly called Urca Processes, reactions 3.2 and 3.3 are particularly known as the Direct Urca process. The name Urca actually refers to a Casino in Rio de Janeiro, and was compared by Gamow to N.S. cooling in the sense that the casino was a perfect “sink” for money, just as neutrino emission is the perfect sink for the N.S.s thermal energy [3]. Neutrino emmision can be a good cooling mechanism because, since these do not interact “much”, they can escape the N.S. quite easily. The Direct Urca reactions are not the only possible neutrino emmiting processes, if for example a meson condensate were present in the N.S. 3 , reactions such as n + π − → n + e + νe. (3.5). n + k − → n + e + νe ,. (3.6). or and their accompanying electron capture reactions similar to eq. ( 3.3) would occur. In the case that free quarks actually exist (Quark-Gluon Plasma), we may have a Quark Direct Urca process such as 2 3. White dwarfs (R ∼ 100km) actually cool via photon emission. See section 1.2.1.

(29) CHAPTER 3. COOLING. 27. Figure 3.1: The figure shows a Feynmann diagram for reaction ( 3.2). During the beta decay of a neutron (udd), a “d” quark is turned unto a “u” quark via the emission of a “W”, and thus a proton (uud) is formed. The other quarks are just spectators that go along for the ride.. u + e → d + νe. (3.7). d → u + e + νe ,. (3.8). and. may also be possible [3]. Each of these cooling mechanisms has its own characteristic cooling curve (T time), so the N.S’s cooling curve depends on the type of matter present.. 3.2. The Direct Urca Threshhold. Let’s look at the Direct Urca process (Eqs. 3.2 and 3.3) a bit more carefully. For this reaction to occur, all particles participating in it must have energies close to their Fermi surface, i.e. they must all have energies of the order of K B Tf . For example, if the neutron were to decay, the states of the final proton and electron would have to be empty, these empty states are more likely to be at energies of the order of K B Tf . If we impose this condition on the neutron decay we obtain the following restriction on the Fermi momenta [3]: P (p)f + P (e)f ≥ P (n)f ,. (3.9). assuming that the momentum of the neutrino is negligible. Recalling that the particle concentration and their Fermi momenta are related by eq. (1.10) n(i) =. P (i)3f 3π 2 h̄3. .. (3.10).

(30) CHAPTER 3. COOLING. 28. So we can say, because of charge neutrality that P (p) f = P (e)f , and thus rewrite eq. ( 3.9) as 2P (p)f ≥ P (n)f. (3.11). and using eq. ( 3.10) we can find a restriction on the proton concentration n(p) ≥. n(n) . 8. (3.12). If we now define the proton fraction as χ=. n(p) , n(n) + n(p). (3.13). we find that [7] 1 ' 11%. (3.14) 9 We have found a minimum proton fraction necessary for momentum to be conserved and for the Direct Urca process to occur. Because of this minimum proton fraction, the Direct Urca process was actually not accepted in the physics community for many years on the basis that the minimum proton fraction was considered to be too high [3]. The proton concentration is very sensitive to the eq. of state, but current nuclear matter models predict that it is possible to have χ ≥ 1/9 [3, 7]. χ≥. 3.3. The Modified Urca Process. If the proton fraction is less than 11%, the Direct Urca process is not possible, and a new particle must be introduced into inequality 3.9 so that momentum can be conserved. This new cooling mechanism can be written as n + (n, p) → (n, p)0 + p + e + νe. (3.15). p + e + (n, p) → (n, p)0 + n + νe. (3.16). and. This cooling mechanism is called the Modified Urca Process. Here, a bystander neutron or proton participates to allow momentum conservation (By absorbing momentum.). This is a second order process and is thus less likely to occur compared to the Direct Urca process. For this reason, a neutron star with a proton fraction less than 11% will cool much more slowly [8]..

(31) CHAPTER 3. COOLING. 3.4. 29. Cooling curves: An outline of the problem. The most important objective of this document is to calculate a temperature-time curve for a N.S. containing ordinary matter. For this, we must make the following assumptions: • The N.S. is composed of ordinary matter (neutrons, protons and electrons), there is no exotic matter present. This could only be justified by comparing the final cooling curve with observations. Also, no superfluid phases will be considered; eventhough these might play an important role in the cooling of N.S.s, they are beyond the scope of this document. • Since the core of the N.S. constitutes about 99% of the mass of the star, the Crust will not be considered (at least in the Neutrino Cooling modes). • The N.S. may be considered as being highly degenerate. This can be justified because temperatures in N.S.s are much less than the Fermi temperature (∼ 1012 K)4 . Fermi-Dirac statistics will be used to describe the N.S. • The N.S. is roughly isothermal. This as a consequence of it being highly degenerate, i.e. the mean free path of a particle in a degenerate fluid is usually very long. More assumptions will be made along the way, but these are probably the most important ones.. 3.4.1. Procedure. In short, the procedure to calculate a cooling curve will be the following: • First calculate the energy loss per unit time (Also called Luminosity or Emissivity.) due to neutrino and photon emission. These will be labeled Ėν and Ėγ respectively. These luminosities as we shall see can be calculated using non relativistic quantum mechanics, in particular we will use the Fermi Golden rule of time dependent perturbation theory. • Noting that the total luminosity (Photons plus neutrinos) is simply Ė = Ėν + Ėγ = −. dE , dt. (3.17). and that dE dt dE = = Cv , dT dt dT 4. See section 1.1.1. (3.18).

(32) CHAPTER 3. COOLING. 30. Cv being the Specific heat capacity at constant volume for the N.S.; we can finally write −(Ėν + Ėγ ) = Cv. dT , dt. (3.19). which can be called the Cooling equation. The next step consists in calculating the Specific heat capacity of the N.S., which can actually be found using statistical Mechanics, assuming that the star is made primarily of neutrons. • Equation 3.19, simple as it may be, governs the cooling of the N.S., so now the final step is to solve this differential equation to obtain a temperature-time curve T (t). The same procedure will be done for both the Direct Urca and the Modified Urca processes, so that both cooling curves can be compared with each other and with actual observations.. 3.5. Direct Urca Luminosity. The neutrino emisivity for the Direct Urca process shall now be calculated. (n → p + e + ν). The rate at which neutrons decay in a netron star can be calculated using the Fermi Golden rule. Qualitatively this can be written as Γν ∝. 2π | < f |Ĥ|i > |2 ∗ (P hase − Space). h̄. (3.20). 5. Were the Bra-Ket is just the transition probability from an initial state to a final state and Ĥ is the weak interaction Hamiltonian. The phase space is the one available for the transition. Note that since a neutron star is a degenerate fermi fluid, the phase space is severely restricted6 .The phase space is reduced because the probability that the final states with energies Ep and Ee are empty, is simply 1 minus the probability of them being occupied. This means that the terms (1 − f p )(1 − fe ) have to 7 be included in equation ( 3.20) . The fi s are simply the Fermi-Dirac distribution functions fi =. 1 Ei −µi kB T. .. (3.21). 1+e An extra term fn has to be included because we have to take into account that the initial neutron with energy En actually exists. If we now want to calculate a luminosity, or energy loss, we can just use eq. 3.20 to write 5. The Fermi golden rule is just an aplication Time-dependent perturbation theory to first order and is treated in Appendix A. 6 This in contrast to the free neutron decay, were any final state is un-occupied 7 Also called Pauli blocking factors..

(33) CHAPTER 3. COOLING. 2π E˙ν = 2 h̄. X. 31. | < f |Ĥ|i > |2 Eν fn (1 − fp )(1 − fe ) δ 4 (P~n − P~p − P~e − P~ν ),. (3.22). were the sum is made over all possible four-momenta. A factor of 2 has been added because the other Direct Urca reaction (e + p → n + ν) has to be considered with equal probability (It’s basically the same Feynmann diagram.). The delta function simply enforces consevarion of energy and momentum. The Weak Interaction Matrix element The matrix element strores the information of the strength of the interaction. When Fermi developed his Golden rule, he stored the information of the strength of the interaction in the coupling constant G f and took the matrix element to be [10] Gf . (3.23) V Here the volume V is the normalization volume of the free particle states, i.e. ~ |Ψi (~r)i) = √1V eik.~r . At the time this was done there was no knoledge of W bosons, so Fermi considered the inteaction potential as being a contact potential. A small correction to the matrix element is motivated by the more recent G.W.S. electroweak theory so that the matrix element becomes [3] < f |Ĥ|i >=. 1 2 G cos2 θc (1 + ga2 ). , (3.24) V2 F This expression reminds us of the weak coupling vertex factor, were there is an axial and a vector coupling. In the previous equation, θ c is the Cabibo angle and ga is the Axial-Vector coupling constant. Hi f =. The Phase Space Factor Using the expression for the matrix element, eq. ( 3.22) becomes E˙ν. 2π 2 GF cos2 θc (1 + ga2 ) h̄ X ∗ Eν fn (1 − fp )(1 − fe ) δ 4 (P~n − P~p − P~e − P~ν ) 2π 2 G cos2 θc (1 + ga2 )P , = h̄ F. = 2. (3.25). were P is defined by P=. X. Eν fn (1 − fp )(1 − fe ) δ 4 (P~n − P~p − P~e − P~ν ) ,. and is usually called The Phase-Space factor.. (3.26).

(34) CHAPTER 3. COOLING. 32. The problem of calculating the emisivity is basically reduced to calculating P. If the reader is just interested in the final result, he (or she) may skip to eq. ( 3.60) We can now change the sum to an integral over three momenta (~ p i ) by introducing the density of states. For illustrative purposes we can first define a quantity P 0 that corresponds to the Phase-space factor available for the Direct Urca process assuming that the initial neutron has a definite four-momentum.. P0 =. Z. V d p ~p 3 h 3. Z. Z. V d p~e 3 h 3. d3 p ~ν. V (1 − fp )(1 − fe ) δ 4 (P~n − P~p − P~e − P~ν ) (3.27) h3. If we now integrate over all possible four-momenta of the neutron, we get. P=. Z. d3 p~n P0 =. V3 h9. Z. d3 p ~n d3 p~p d3 p~e d3 p ~ν fn (1 − fp )(1 − fe )δ 4 (P~n − P~p − P~e − P~ν ). (3.28) If we now use spherical coordinates, we can separate P into a radial part and an angular part. The angular part A can be written as: A=. Z. dΩn dΩp dΩe dΩν δ 3 (p~n − p~p − p~e − p~ν ).. (3.29). pp + p~e |, we can rewrite the delta as δ 3 (~ pn − p ~p − p~e ). Using the fact that |~ pν | |~ We can now use a property of the delta function: δ 3 (~ pn − p~p − p ~e ) = δ(|~ pn | − |~ pp + p~e |)δ(Ωn − Ωe−p ) Integrating over Ωn we get, A=. Z. dΩp dΩe dΩν δ(pn − |~ pp + p~e |). 1 . |~ p n |2. 1 p2n. (3.30). (3.31). Noting that δ(pn − |~ pp + p~e |) = δ(pn − (p2e + p2p + 2pe pp cos θ)1/2 ) = δ(f (cos θ)) and using δ(f (cos θ)) =. 1 f 0 (cos a). δ(cos θ − a). (3.32). (3.33). were f (a) = 0. The derivative f 0 can now be expressed as f 0 (cos θ) =. pe pp pe pp = pn (p2e + p2p + 2pe pp cos θ)1/2. (3.34).

(35) CHAPTER 3. COOLING. 33. So, δ(f (cos θ)) = δ(cos θ − a). pn . pe pp. (3.35). Now we can rewrite the angular part A as Z 1 δ(cos θ − a). A = dΩp dΩe dΩν pe pp pn. (3.36). Noticing that the delta function fixes θ, when we integrate over solid angle Ω p or Ωe we will obtain a factor of 2π instead of the usual 4π. So A=. 32π 2 2π(4π)2 = pe pp pn pe pp pn. (3.37). And Z 32π 2 V 3 P= pe pp pn pν2 Eν dpe dpp dpn dpν fn (1 − fp )(1 − fe )δ(En − Ep − Ee − Eν ). h9 (3.38) We would like to integrate over energy instead of momentum. We can aproximate the protons and neutrons as being non relativistic particles [2]. pi dpi f or i = n, p (3.39) mi This aproximation can be justified by racalling that at typical N.S. temperatures (∼ 109 K) we have dEi =. kB (109 )◦ K ∼ 10−14 J ∼ mn vn2 ⇒ vn ∼ 106 m/s ∼ 0.001c. On the other hand, we can take the electrons as being highly relativistic in tipical N.S. temperatures, so Ee = pe c ⇒ dEe = dpe c For neutrinos it’s always valid to write dEν = dpν c. So we can now write P in terms of energy. Thus P. Z . . =. 32π 3 V 3 h9. ∗. fn (1 − fp )(1 − fe )δ(En − Ep − Ee − Eν ). Ee dEe c2. (dEp mp )(dEn mn ). Eν3 dEν c3. !. (3.40). Since the electrons are highly degerenate and only energies close to the fermi surface are considered, we may simply repalce E e by the fermi energy µe and remove it from the integral8 . 8 We cannot do the same with the neutrino since these are not degenerate, this is because they can escape the N.S. quite easily. Degenerate neutrinos are only thought to exist for a short period of time durig a supernova explosion when the opacity is extremely high..

(36) CHAPTER 3. COOLING. 34. Z. 32π 3 V 3 mp mn µe dEe dEp dEn Eν3 dEν P = h9 c5 ∗ fn (1 − fp )(1 − fe )δ(En − Ep − Ee − Eν ) = S. Z. (3.41). dEe dEp dEn Eν3 dEν fn (1 − fp )(1 − fe )δ(En − Ep − Ee − Eν ),. Were S is defined by 32π 3 V 3 mp mn µe . h9 c5 Now we make the following changes of variable: S=. Eν =. (3.42). Eν kT. En − µ n kT −Ee + µe Ee = kT −Ep + µp (3.43) Ep = kT Recalling the chemical equilibrium condition (µ n = µp + µe ), the delta function can be written as En =. 1 δ(En + Ep + Ee − Eν ) kT Now P becomes P = S(kT ). 6. .. Z. Eν3 δ(En + Ep + Ee − Eν ). dEe dEp dEn E E p n 1 + e 1 + e 1 + e Ee. (3.44). (3.45). Recalling that the delta function is just the Fourier transform of a constant, we can use the representation 1 δ(En + Ep + Ee − Eν ) = 2π So that S P= 2π S = 2π. Z. Z. Eν3. Eν3. dEν. dEν. Z. Z. Z. eiz(En +Ep +Ee −Eν ) dz. (3.46). eiz(En +Ep +Ee −Eν ) dz. dEe dEp dEn E E p n 1 + e 1 + e 1 + e Ee. (3.47). eiz(En +Ep +Ee e−Eν ) dz. dEp dEe dEn E E p n 1 + e 1 + e 1 + e Ee. (3.48).

(37) CHAPTER 3. COOLING. 35. y (Im). 6 q. . 2π. 6. ?. x (Re) -. -. Figure 3.2: Path taken for integral 3.51. The vertical segments do not contribute to the integral when one integrates from −∞ to ∞. We can first calculate Z. ∞. dEi 1 + e Ei −∞ by considering the closed contour integral in the complex plane I. eizEi. (3.49). dw . (3.50) 1 + ew Were w is complex (w = x + iy). Note that this contour integral has a pole at w = iπ. Making a direct calculation for the residues, we find that the residue is −e−πz . The most convenient contour for the calculation of eq. ( 3.49) is a closed contour that passes through 2π in the imagianry axis, and of course, it should pass through the real axis as shown in figure 3.2. eizw. Now using the Residue Theorem. I. e =. izw. Z. dw .= 1 + ew. ∞ −∞. Z. ∞ −∞. eizx dx − 1 + ex. eizx dx − e2πz 1 + ex. Z. ∞ −∞. Z. ∞ −∞. eiz(2πi+x) dx 1 + e2πi+x. eizx dx = −2πie−πz 1 + ex. (3.51) (3.52). And the definite integral ( 3.49) becomes Z. ∞ −∞. eizEi. 2πie−πz dEi = . 1 + e Ei 1 − e−2πz. (3.53). We can now rewrite eq ( 3.48) as. S P= 2π. Z. ∞ 0. Eν3. dEν. Z. ∞ −∞. e. −izEν. 2πie−πz 1 − e−2πz. !3. dz.. (3.54).

(38) CHAPTER 3. COOLING. 36. y (Im) 6. . 1. 6. - q -. ?. - -x (Re). Figure 3.3: Path taken for integral 3.55 Once again we can make a contour integral to calculate the second integral in eq. ( 3.54). Note that this integral has a pole at z = 0. In this case a suitable contour is one that passes though 1 in the imaginary axis as shown in the figure.. I Z. ∞ −∞. e. −ixEν. e. 2πie−πx 1 − e2πx. =. Z. ∞ −∞. e. 2πie−πz 1 − e2πz. −izEν. !3. dz −. −ixEν.  . Z. ∞ −∞. e. . dz =. −iEν (x+i). 2πie−πx 1 − e2πx. = −2πiRes e−ixEν. !3. !3. . (3.55). 2πie−π(x+i) 1 − e−2π(x+i). !3. dz. . 1 + eEν dz. 2πie−πx 1 − e2πx. !3   . .. (3.56). A calculation of the Residue can be made by making a Laurent expansion of the expression inside the residue brackets. This expansion gives x i(π 2 + x2 ) 1 + − . z3 z2 2z. (3.57). This means that the residue is 12 (π 2 + x2 ). Finally eq. ( 3.54) can be exressed as i P= S 2. Z. ∞ 0. . Eν3 1 + eEν. −1. (π 2 + x2 ). (3.58). 457π 6 . (3.59) 5040 Recalling eq. ( 3.42) for S, we can finally express the neutrino luminosity (eq. 3.26) for the Direct Urca process as P =S.

(39) CHAPTER 3. COOLING. 37. 457π V mp mn µe E˙ν = (kT )6 G2f cos2 θc (1 + ga2 ) J/s (3.60) 1080 h̄10 c5 This is the energy loss via neutrino emision, it coincides with the result obtained by Pethick [3]. The (kT )6 dependence is particular of the Direct Urca process as we shall see. The previous result, together with the specific heat, will allow us to calculate a cooling curve for neutrino cooling.. 3.6. Modified Urca Luminosity. The Luminosity for the Modified Urca process shall now be calculated (n 1 + n1 → n02 + p + e + ν. We shall follow a similar procedure to Shapiro [2]). The procedure followed is the same as in the Direct Urca process, only that now there is a bystander particle involved. Here we assume that the bystander particle is a neutron. We could also assume that the bystander is a proton, which would give us a different luminosity, but since there a more neutrons than protons, only the neutron brach shall be considered. Also, even if the proton concentration would be high enough for us to consider a proton branch, the Direct Urca process would occur and would be much more effective than the Modified Urca process as we shall see. The luminosity can be written as Ėν. =. 2X 2π Eν δ 4 (P~n1 + P~n2 − P~n02 − P~p − P~e − P~ν ) < f |Ĥ|i > h̄ P i. ∗. fn1 fn2 (1 − fn02 )(1 − fp )(1 − fe ).. (3.61). Here, the matrix element again refers to a transition probability, the P i refer to the particles four-momenta, and the fi refer to the Fermi-Dirac distributions. The matrix element however, is not as simple as the one used for beta decay (Direct Urca process), since there is now a bystander particle that interacts via the strong interaction. The Weak Interaction Matrix element The strong interaction can in this case, be modeled as a one pion (π) exchange between particles. This model is called the One Pion Exchange model (OPE) 9 , a result for the matrix element was obtained by Yakovlev and Kaminker [11] and is written as | < f |Ĥ|i > |2 =. 16G2f cos2 θc ga2 m4π Ef2e. (3.62). Let us first calculate the phase space factor for a definite energy for neutrons n 1 and n2 (Just for illustrative purposes). 9. The bystander particle interacts via the Strong force, which can be modeled by an effective theory that considers a Pion exchange. This was first done by Yukawa [10]..

(40) CHAPTER 3. COOLING. Z. 38. Z. Z. V 3~ V 3~ V 3~ d Pp 3 d Pe d Pν Eν δ 4 (P~f − P~i ) 3 h h h3 fn1 fn2 (1 − fn02 )(1 − fp )(1 − fe ).. P0 = ∗. d3 P~n02. (3.63). The Phase Space Factor If we now integrate over all possible four-momenta available for neutrons n 1 and n2 , we get P. = = ∗. Z. 3~. d P n1 Z. Z. d3 P~n2 P 0. V4 d3 P~n1 d3 P~n2 d3 P~p d3 P~e d3 P~ν Eν δ 4 (P~f − P~i ) h12 fn1 fn2 (1 − fn02 )(1 − fp )(1 − fe ).. (3.64). The problem is again reduced to calculating the previous phase space factor. If the reader is just interested in the final result, he (or she) may skip to eq. ( 3.96). Using spherical coordinates, we can first integrate the angular part of P. A=. Z. dΩn1 dΩn2 dΩp dΩe dΩν δ 3 (P~f − P~i ).. (3.65). If we now note that the four-momentum of the neutrino is small, i.e. P~ν P~p + P~e + P~n02 ,. (3.66). so the delta function can now be written as δ 3 (P~f − P~i ) ' δ(P~n1 + P~n2 − P~n02 − P~p − P~e ).. (3.67). Now we can exploit the fact that the delta function can be written as δ(P~n1 + P~n2 − P~n02 − P~p − P~e ) =. δ(|P~n1 | + |P~n2 − P~n02 − P~p − P~e |) δ(Ωn1 − Ω−n2 +n02 +p+e ) P12. n1. ,. (3.68). were the Pi are the magnitudes of the four-momenta. Now integrating over Ωn1 we obtain A=. Z. dΩn2 dΩn02 dΩp dΩe dΩν. 1 δ(|P~n1 | + |P~n2 − P~n02 − P~p − P~e |). Pn21. We can once again rewrite the delta function as. (3.69).

(41) CHAPTER 3. COOLING. 39. δ(|P~n1 | + |P~n2 − P~n02 − P~p − P~e |) = δ(|P~n1 | − |P~n2 + P~R |),. (3.70). were R stands for “the Rest of the particles ”. Now exploiting the delta identity the previous equation can be written as = δ(Pn1 − (Pn20 + PR2 + 2Pn02 PR cos θ)1/2 ) = δ(f (cos θ)) 2. =. 1 δ(cos θ − a), f 0 (a). (3.71). were f 0 (cos theta) just refers to the derivative and ’a’ is the value at which f (a) = 0.. f 0 (cos θ) =. Pn02 PR (Pn20 + PR2 + 2Pn02 PR cos θ)1/2 2. =. Pn02 PR. ⇒ δ(f (cos θ)) = A=. Z. (3.72). P n1 P n1 δ(cos θ − a) Pn02 PR. dΩn2 dΩn02 dΩp dΩe dΩν. 1 δ(cos θ − a) Pn02 PR Pn1. (3.73) (3.74). Now recalling that θ is the angle between P~n02 and P~R = P~p + P~e + P~n2 , we can note that there is a restriction over the solid angle. This means that when we integrate over Ωn02 , the result will be 2π instead of the usual 4π. So integrating over all solid angles we obtain A=. 2π(4π)4 . Pn02 |P~p + P~e − P~n2 |Pn2. (3.75). If we now remember that the particles involved in the reaction have energies close to the Fermi surface, we can write |P~p + P~e − P~n2 | ' Pn2 ,. (3.76). simply because the neutron concentration is much greater than the proton and electron concentration10 . We now have for the angular part of the phase factor A= 10. (4π)5 , 2Pn02 Pn1 Pn2. Remember that the Fermi momentum is proportional to the concentration.. (3.77).

(42) CHAPTER 3. COOLING. 40. so the phase factor P (Eq. ( 3.64)) can now be written as (In spherical coordinates). P. Z. (4π)5 V = Pn1 dPn1 Pn2 dPn2 Pn02 dPn02 Pp2 dPp Pe2 dPe Pν2 dPν Eν 2h10 ∗ δ(En1 + En2 − En02 − Ep − Ee − Eν ) ∗. fn1 fn2 (1 − fn02 )(1 − fp )(1 − fe ).. (3.78). To integrate over energy instead of momentum, we can approximate the protons and neutrons as being non relativistic particles (Just as we did with the direct Urca process.). dEi =. Pi dpi mi. f or i = n, p. (3.79). We can also again take the electrons as being highly relativistic Ee = Pe c ⇒ dEe = dPe c. (3.80). For neutrinos we take as usual dEν = dPν . So now P becomes P = ∗. (4π)5 V mn1 mn2 mn02 mp Z. = S. 2h10 dEn1 dEn2 dEn02 dEp Z. 3 Y Ee2 Eν (1 − fi ) δ dE dE f f e 3 ν n1 n2 c3 c i=1. dEn1 dEn2 dEn02 dEp Ee2 c3 dEe Eν c3 dEν fn1 fn2. 3 Y. i=1. (1 − fi ) δ,. (3.81). were S si defined as11 (4π)5 V m3n mp Pfp µ2e . 2h10 c6 Now we make the following changes of variable S=. E n1. =. E n2. =. En02. =. (3.82). E n1 − µ n1 kT E n2 − µ n2 kT −En2 + µn02 kT. 11 Once again we have taken the electrons to be highly degenerate, so only energies close to the Fermi surface are considered. This means that we replace Ee by the Fermi energy µe which is constant..

(43) CHAPTER 3. COOLING. 41 −Ep + µp kT −Ee + µe kT Eν kT. Ep = Ee = Eν. =. (3.83) (3.84). Recalling the chemical equilibrium condition, the delta function can be written as 1 δ(En1 + En2 + En02 + Ep + Ee − Eν ), kT. (3.85). so now P becomes P = S(kT )8. Z. Eν δ(En1 + En2 + En02 + Ep + Ee − Eν ). 5 Y. dEi . Ei 1 + e i=1. (3.86). Again, recalling that the delta function is the Fourier transform of a constant, we can write δ. 5 X i=1. Ei − Eν. !. Z. 1 = 2π. (. exp iz. 5 X i=1. Ei − Eν. !). dz,. (3.87). dEi 1 + e Ei i=1. (3.88). so that S P= 2π. Z. Eν3. dEν. Z. (. 5 X. exp iz. i=1. Ei − Eν. !). dz. 5 Y. We can first calculate Z. ∞ −∞. eizEi. dEi , 1 + e Ei. (3.89). which was already calculated in section (3.5). By using the residue theorem one obtains. We can now write eq. ( 3.88) as S P= 2π. Z. ∞ 0. Eν3. dEν. 2πie−πz . 1 − e−2πz Z. ∞ −∞. e. −izEν. (3.90). 2πie−πz 1 − e−2πz. !5. dz.. (3.91). Again using the residue theorem, we can calculate the second integral in eq. ( 3.91). Making a contour integral around the path shown in figure ( 3.4) we get.

(44) CHAPTER 3. COOLING. 42. y (Im) 6 . . 6. qiπ. ?. x (Re) -. -. Figure 3.4: Path taken for integral ( 3.91). The vertical segments do not contribute to the integral when one integrates from −∞ to ∞.. I Z. ∞ −∞. e. −ixEν. 2πie−πx 1 − e2πx Z. =. ∞ −∞. e. 2πie−πz 1 − e2πz. −izEν. !5. dz −. e−ixEν  . Z. ∞ −∞. e. Making a Laurent expansion of. . 2πie−πx 1 − e2πx. 2πie−πx 1−e2πx. dz =. −iEν (x+i). !5. . 5. 2πie−π(x+i) 1 − e−2π(x+i). !5. dz. . (3.92). .. (3.93). 1 + eEν dz. 2πie−πx 1 − e2πx. = −2πiRes e−izEν . !5. we find. !5   . −i Eν2 i(4π 2 + 3Eν2 ) 5π 2 Eν2 + Eν3 i(9π 4 + 10π 2 Eν2 + Eν4 ) − 4 + + − + ... z5 z 6z 3 6z 2 24z so that the residue is. P = =. −i 4 24 (9π. (3.94). + 10π 2 Eν2 + Eν4 ) and eq ( 3.91) becomes. Z. S(kT )8 ∞ 3 −i (9π 4 + 10π 2 Eν2 + Eν4 ) 1 + eEν dEν Eν 2π 24 0 8 8 11513π 7 S(kT ) 11513π =S (kT )8 . 2π 120960 241920. (3.95). Recalling the definition of S (Eq. ( 3.82)), and the expression for the weak matrix element (Eq. ( 3.62)), we can finally we can express Ėν eq. ( 3.61) as Ėν =. 11513 G2F cos2 θc ga2 V m3n mp Pfp (kT )8 60480πh̄ 10 c6 m4π. (3.96).

(45) CHAPTER 3. COOLING. 43. This is the Luminosity for neutrino emission for the Modified Urca process. Notice the (kT )8 dependence, in contrast to the (kT ) 6 dependence for the Direct Urca process. The Fermi momentum and energy terms contained in the constant depend on the concentration of protons and electrons respectively, so we expect there to be slight variations in Luminosity of N.S.s with different densities. As we shall see, this luminosity produces a much slower cooling than the Direct Urca process. We can now proceed to calculate a cooling curve, using the specific heat for a N.S... 3.7. Specific Heat Capacity. In this section we shall calculate the specific heat capacity for a N.S., assuming for simplicity that it is composed mainly of degenerate neutrons. This is a standard calculation for electrons and can be found in many references, here we shall rely mainly on Callen [16]. Recall that the specific heat capacity is given by Cv =. . ∂U ∂T. . ,. (3.97). V. were U (T, V, µ) is the internal energy, the quantity that we shall first calculate 12 . The internal energy can be calculated using the Fermi-Dirac distribution, and so we can write 2V U= 3 h. Z. d3 p~. E . 1 + exp β(E − µ). (3.98). Here the factor of 2 accounts for the spin of the neutron. Using spherical coordinates we can write. U. = = =. Z. E p2 dp 1 + exp β(E − µ) √ Z 8πV 2mEmEdE h3 1 + exp β(E − µ) √ Z E 3/2 dE 8πV m 2m . h3 1 + exp β(E − µ) 8πV h3. (3.99). The previous integral can be calculated using the Sommerfeld expansion 13 [16] 12. Recall that the internal energy was for the ground state (T = 0) of a N.S. was already calculated in section 1.3 13. Z. ∞ 0. φ(E)dE = 1 + exp β(E − µ). Z. µ. φ(E)dE + 0. π2 (kT )2 φ0 (µ) + ... 6. (3.100).